Un-Weyl-ing the Clifford Hierarchy

Abstract

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

arXiv commentLicense changed. Accepted to Quantum on 26 Nov 2020

References in corpus (13)

• Quantum Teleportation is a Universal Computational Primitive
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• The Resource Theory of Stabilizer Computation
• Discrete phase space based on finite fields
• Magic state distillation with low overhead
• Methodology for quantum logic gate constructions
• Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
• Quantum error correction with only two extra qubits
• Fault-tolerant quantum computation with few qubits
• Diagonal gates in the Clifford hierarchy
• Semi-Clifford operations, structure of ${\mathcal{C}}_k$ hierarchy, and gate complexity for fault-tolerant quantum computation
• Flag fault-tolerant error correction, measurement, and quantum computation for cyclic CSS codes
• Unifying the Clifford Hierarchy via Symmetric Matrices over Rings

Cited by in corpus (7)

• Efficient quantum gate teleportation in higher dimensions
• Designing the Quantum Channels Induced by Diagonal Gates
• Decomposition of Clifford Gates
• Scaling W state circuits in the qudit Clifford hierarchy
• Blindly Factorizing 21 Quantumly
• Targeted Clifford logical gates for hypergraph product codes
• Characterization of non-adaptive Clifford channels